two, compareth not two with four, nor with six, for two is the name of a quantity absolute. But he that answereth it is more by half of four, or by a third of six, compareth the difference with one of the differing quantities. For halfs and thirds, &c. are names of quantity compared. From hence there ariseth two species or kinds of (_ratio_) proportion, into which the general word _proportion_ may be divided. The one whereof, namely, that wherein the difference is not compared with either of the differing quantities, is called _ratio arithmetica_, arithmetical proportion; the other _ratio geometrica_, geometrical proportion; and, because this latter is only taken notice of by the name of proportion, simply _proportion_. Having considered this, I defined proportion, chapter II. article 3, in this manner: “_Ratio est relatio antecedentis ad consequens secundum magnitudinem_:” _Proportion is the relation of the antecedent to the consequent in magnitude_; having immediately before defined relatives, antecedent, and consequent, in the same article, and by way of explication added, that such relation was nothing else but that one of the quantities was equal to the other, or exceeded it by some quantity, or was by some quantity exceeded by it. And for exemplification of the same, I added further, that the proportion of three to two was, that three exceeded two by a unity; but said not that the unity, or the difference whatsoever it were, was their proportion, _for unity, and to exceed another by unity_, is not the same thing. This is clear enough to others; let us therefore see why it is not so to you. You say I make proportion to consist in that which remaineth after the lesser quantity is subtracted out of the greater; and that you make it to consist in the quotient, when one number is divided by the other. Wherein you are mistaken; first, in that you say, I make the proportion to consist in the remainder. For I make it to consist in the act of exceeding, or of being exceeded, or of being equal; whereas the remainder is always an absolute quantity, and never a proportion. To be more or less than another number by two, is not the number two; likewise to be equal to two, where the difference is _nothing_, is not that _nothing_? Again, you mistake in saying the proportion consisteth in the quotient. For divide twenty by five, the quotient is four. Is it not absurd to say that the proportion of five to twenty, or of twenty to five, is four? You may say the proportion of five to twenty, is the proportion of one to four. And so say I. And you may therefore also say, that the proportion of one to four is a measure of the proportion of five to twenty, as being equal. And so say I. But that is only in geometrical proportion, and not in proportion universally. For though the _species_ obtain the denomination of the genus, yet it is not the _genus_. But as the quotient giveth us a measure of the proportion of